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This paper presents a theoretical analysis of quantum-mechanical systems on elliptical paths, comparing non-Hermitian and Hermitian Hamiltonian models to explore spectral degeneracy and symmetry properties. The research contributes fundamental insights into exactly solvable Schrödinger equations but does not address practical applications in quantum computing hardware, communication protocols, or algorithmic advancements.
Two simple models derived from a quantum-mechanical particle on an elliptical path
arXiv:2512.09905v3 Announce Type: replace Abstract: We analyze two simple models derived from a quantum-mechanical particle on an elliptical path. The first Hamiltonian operator is non-Hermitian but equivalent to an Hermitian operator. It exhibits the same two-fold degeneracy as the particle on a circular path. More precisely, the spectrum is $E_{n}=n^{2}E_{1},\ n=0,\pm 1,\pm 2,\ldots $, $E_{1}>0$. The second Hamiltonian operator is Hermitian and does not exhibit such degeneracy. In this case the nth excited energy level splits at the nth order of perturbation theory. Both models can be described in terms of symmetry point groups with one-dimensional irreducible representations but the non-Hermitian example exhibits an additional symmetry that explains the two-fold degeneracy. The Schr\"{o}dinger equation for the first example is exactly solvable.
Abstract:We analyze two simple models derived from a quantum-mechanical particle on an elliptical path. The first Hamiltonian operator is non-Hermitian but equivalent to an Hermitian operator. It exhibits the same two-fold degeneracy as the particle on a circular path. More precisely, the spectrum is $E_{n}=n^{2}E_{1},\ n=0,\pm 1,\pm 2,\ldots $, $E_{1}>0$. The second Hamiltonian operator is Hermitian and does not exhibit such degeneracy. In this case the nth excited energy level splits at the nth order of perturbation theory. Both models can be described in terms of symmetry point groups with one-dimensional irreducible representations but the non-Hermitian example exhibits an additional symmetry that explains the two-fold degeneracy. The Schrödinger equation for the first example is exactly solvable.
arXiv:2512.09905 [quant-ph]
(or arXiv:2512.09905v3 [quant-ph] for this version)
From: Francisco Fernández Dr. [view email] [v1] Wed, 10 Dec 2025 18:33:49 UTC (12 KB) [v2] Sun, 15 Mar 2026 13:24:31 UTC (12 KB) [v3] Wed, 9 Sep 2026 21:18:09 UTC (13 KB)
Full text under the publisher's open licence · 157 words
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